The function is .

Direct comparison test:

Let for all .

1. If converges, then converges.

2. If diverges, then diverges.

Consider the series .

is compared with .

.

Consider .

The series is in the form of -series.

The -series is converges if and only if .

Compare with -series.

Here .

The series is converges.

Therefore, is converges.

converges for all values of .

Apply derivative on each side with respect to .

.

The summation form is .

.

When , .

, when .

Consider .

The series is in the form of -series.

The -series is converges if and only if .

Compare with -series.

Here .

The series is diverges.

diverges when ,  is an integer.

Consider .

Apply derivative on each side with respect to .

.

.

The sine function is oscillates between .

The series is converges only when .

is zero when is multiples of .

Therefore, , is an integer.

converges when .

converges for all values of .

diverges when ,  is an integer.

converges when .